相关论文: Z4-linear Hadamard and extended perfect codes

A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the…

Studying binary perfect codes we show the existence of homogeneous nontransitive codes. Thus, as far as perfect codes are concerned, the propelinear codes are strictly contained in transitive codes, wheresas homogeneous codes form a strict…

In this paper we obtain the necessary condition for the existence of perfect $k$-colorings (equitable $k$-partitions) in Hamming graphs $H(n,q)$, where $q=2,3,4$ and Doob graphs $D(m,n)$. As an application, we prove the non-existence of…

All generalized Hadamard matrices of order 18 over a group of order 3, H(6,3), are enumerated in two different ways: once, as class regular symmetric (6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a group of…

We update the list of odd integers n<10000 for which an Hadamard matrix of order 4n is known to exist. We also exhibit the first example of base sequences BS(40,39). Consequently, there exist T-sequences TS(n) of length n=79. The first…

We construct orthogonal arrays OA$_{\lambda} (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / \lambda$ is as large as possible; these…

We computationally resolve an open problem concerning the expressibility of $4 \times 4$ full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over $\mathbb{F}_2$, we identify 5,304 counterexamples among…

Using reversible Hadamard difference sets, we construct symmetric Bush-type Hadamard matrices of order $4m^4$ for all odd integer $m$.

We prove that the class of $\Z_2\Z_2[u]$-linear codes is exactly the class of $\Z_2$-linear codes with automorphism group of even order. Using this characterization, we give examples of known codes, e.g. perfect codes, which has a…

Using integer linear programming and table-lookups we prove that there is no binary linear $[1988, 12, 992]$ code. As a by-product, the non-existence of binary linear codes with the parameters $[324, 10, 160]$, $[356, 10, 176]$,…

A binary code of blocklength $n$ and codebook size $M$ is called an $(n,M)$ code, which is studied for memoryless binary symmetric channels (BSCs) with the maximum likelihood (ML) decoding. For any $n \geq 2$, some optimal codes among the…

Linear codes with complementary-duals (LCD) are linear codes that intersect with their dual trivially. Multinegacirculant codes of index $2$ that are LCD are characterized algebraically and some good codes are found in this family. Exact…

In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and…

All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact…

We show that if a Barker sequence of length $n>13$ exists, then either $n=3979201339721749133016171583224100$, or $n > 4\cdot10^{33}$. This improves the lower bound on the length of a long Barker sequence by a factor of nearly 2000. We also…

We solve several first questions in the table of small parameters of completely regular (CR) codes in Hamming graphs $H(n,q)$. The most uplifting result is the existence of a $\{13,6,1;1,6,9\}$-CR code in $H(n,2)$, $n\ge 13$. We also…

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of…

This paper aims to construct optimal quaternary additive codes with non-integer dimensions. Firstly, we propose combinatorial constructions of quaternary additive constant-weight codes, alongside additive generalized anticode construction.…

We give a complete classification of binary linear complementary dual codes of lengths up to $13$ and ternary linear complementary dual codes of lengths up to $10$.

It is shown that the residue code of a self-dual $\mathbb{Z}_4$-code of length $24k$ (resp.\ $24k+8$) and minimum Lee weight $8k+4 \text{ or }8k+2$ (resp.\ $8k+8 \text{ or }8k+6$) is a binary extremal doubly even self-dual code for every…